Weighted Kaplan-Meier Tests for Umbrella Alternatives

The problem of testing for umbrella alternatives in a one-way layout with right-censored survival data is considered. Testing procedures based on the two-sample weighted Kaplan-Meier statistics suggested by Pepe and Fleming (1989, Biometrics, 45, 497–507; 1991, J. Roy. Statist. Soc. Ser. B, 53, 341–352) are suggested for both cases when the peak of the umbrella is known or unknown. The asymptotic relative efficiency of the weighted Kaplan-Meier test and the weighted logrank test proposed by Chen and Wolfe (2000, Statist. Sinica, 10, 595–612) is computed for the umbrella peak-known setting where the piecewise exponential survival distributions have the proportional or crossing hazards, or the related hazards differ at early or late times. Moreover, the results of a Monte Carlo study are presented to investigate the level and power performances of the umbrella tests. Finally, application of the proposed procedures to an appropriated data set is illustrated.     

Average projection type weighted Cramér- von Mises statistics for testing some distributions

This paper addresses the problem of testing goodness-of-fit for several important multivariate distributions: (I) Uniform distribution on p-dimensional unit sphere; (II) multivariate standard normal distribution; and (III) multivariate normal distribution with unknown mean vector and covariance matrix. The average projection type weighted Cramér-von Mises test statistic as well as estimated and weighted Cramér-von Mises statistics for testing distributions (I), (II) and (III) are constructed via integrating projection direction on the unit sphere, and the asymptotic distributions and the expansions of those test statistics under the null hypothesis are also obtained. Furthermore, the approach of this paper can be applied to testing goodness-of-fit for elliptically contoured distributions.     

Different kinds of sufficiency in the general Gauss-Markov model

Sufficiency is one of the fundamental notions in mathematical statistics. In connection with the general linear Gauss-Markov model GM (y,Xβ, σ ² V), there are some modifications of this notion such as linear sufficiency (Baksalary and Kala, Drygas) invariant linearly sufficiency (Oktaba, Kornacki, Wawrzosek) and quadratic sufficiency (Mueller). All these variants denote such transformations of the model GM that preserve properties essential in statistical inference. In the present paper we give mutual relations between above three classes of statistics. Communicated by Gejza Wimmer     

Distributions of orientations on Stiefel manifolds

The Riemann space whose elements are m × k (m k) matrices X, i.e., orientations, such that X′X = I_k is called the Stiefel manifold V_k,m. The matrix Langevin (or von Mises-Fisher) and matrix Bingham distributions have been suggested as distributions on V_k,m. In this paper, we present some distributional results on V_k,m. Two kinds of decomposition are given of the differential form for the invariant measure on V_k,m, and they are utilized to derive distributions on the component Stiefel manifolds and subspaces of V_k,m for the above-mentioned two distributions. The singular value decomposition of the sum of a random sample from the matrix Langevin distribution gives the maximum likelihood estimators of the population orientations and modal orientation. We derive sampling distributions of matrix statistics including these sample estimators. Furthermore, representations in terms of the Hankel transform and multi-sample distribution theory are briefly discussed.     

On the joint distribution of studentized order statistics

In this paper, the joint distribution of some special linear combinations of the (internally) studentized order statistics are derived for both normal and exponential populations; the exact relationship between their pdf's is also obtained. The exact sampling distributions of studentized extreme deviation statistic, which has been proposed by Pearson and Chandra Sekar (1936,Biometrika,28, 308–320), are derived for these two populations. An application to the most powerful location and scale invariant test is discussed briefly.     

Some estimation theory on the sphere

Summary We consider distributions with densities of the formf(μ′x) andf(‖x _v ‖) where μ andx are unit vectors inR ^q and ‖x _v ‖ is the norm of the part ofx in somes dimensional subspaceV ofR ^q . For several loss functions, optimal Bayesian and Pitman estimators of μ andV are given. When uniform priors are used, these estimators are identical. Then the infinitesimal robustness characteristics of several special cases of these estimators are calculated.     

Limit theorems for additive statistics based on moving average samples

We study statistics based on samples of moving averages generated by stationary sequence of random variables. The central limit theorem (CLT) is proved for sequences of observations defined by an analytic function of moving averages under consideration. For U- and V -statistics with canonical (degenerate) kernels, the limit distributions are studied.     

Comparison of ‘Tail Method’ Exact Confidence Limits for the Difference of Binomial Probabilities

Suppose that Y ₁ and Y ₂ are independent and have Binomial(n ₁,p ₁) and Binomial (n ₂,p ₂) distributions respectively. Also suppose that θ=p ₁−p ₂ is the parameter of interest. We consider the problem of finding an exact confidence limit (either upper or lower) for θ. The solution to this problem is very important for statistical practice in the health and life sciences. The ‘tail method’ provides a solution to this problem. This method finds the exact confidence limit by exact inversion of a hypothesis test based on a specified test statistic. Buehler (J. Am. Stat. Assoc. 52, 482–493, 1957) described, for the first time, a finite-sample optimality property of this confidence limit. Consequently, this confidence limit is sometimes called a Buehler confidence limit. An early tail method confidence limit for θ was described by Santner and Snell (J. Am. Stat. Assoc. 75, 386–394, 1980) who used the maximum likelihood estimator of θ as the test statistic. This confidence limit is known to be very inefficient (see e.g. Cytel Software, StatXact, version 6, vol. 2, 2004). The efficiency of the confidence limit resulting from the tail method depends greatly on the test statistic on which it is based. We use the results of Kabaila (Stat. Probab. Lett. 52, 145–154, 2001) and Kabaila and Lloyd (Aust. New Zealand J. Stat. 46, 463–469, 2004, J. Stat. Plan. Inference 136, 3145–3155, 2006) to provide a detailed explanation for the dependence of this efficiency on the test statistic. We consider test statistics that are estimators, Z-statistics and approximate upper confidence limits. This explanation is used to find the situations in which the tail method exact confidence limits based on test statistics that are estimators or Z-statistics are least efficient.     

High dimensional limit theorems and matrix decompositions on the Stiefel manifold

The main purpose of this paper is to investigate high dimensional limiting behaviors, as m becomes infinite (m → ∞), of matrix statistics on the Stiefel manifold V_{k, m}, which consists of m × k (m ≥ k) matrices X such that X′X = I_k. The results extend those of Watson. Let X be a random matrix on V_{k, m}. We present a matrix decomposition of X as the sum of mutually orthogonal singular value decompositions of the projections P X and P X, where and are each a subspace of R^m of dimension p and their orthogonal compliment, respectively (p ≥ k and m ≥ k + p). Based on this decomposition of X, the invariant measure on V_{k, m} is expressed as the product of the measures on the component subspaces. Some distributions related to these decompositions are obtained for some population distributions on V_{k, m}. We show the limiting normalities, as m → ∞, of some matrix statistics derived from the uniform distribution and the distributions having densities of the general forms f(P X) and f(m^1/2P X) on V_{k, m}. Subsequently, applications of these high dimensional limit theorems are considered in some testing problems.     

M-solid varieties generated by lattices

Following W. Taylor, we define an identity to be hypersatisfied by a variety V iff, whenever the operation symbols of V are replaced by arbitrary terms (of appropriate arity) in the operations of V, then the resulting identity is satisfied by V in the usual sense. Whenever the identity is hypersatisfied by a variety V, we shall say that is a hyperidentity of V, or a V hyperidentity. When the terms being substituted are restricted to a submonoid M of all the possible choices, is called an M-hyperidentity, and a variety V is M-solid if each identity is an M-hyperidentity. In this paper we examine the solid varieties whose identities are lattice M-hyperidentities. The M-solid varieties generated by the variety of lattices in this way provide new insight on the construction and representation of various known classes of non-commutative lattices. Received October 8, 1999; accepted in final form March 22, 2000.     

Concentrated matrix Langevin distributions

This paper concerns the matrix Langevin distributions, exponential-type distributions defined on the two manifolds of our interest, the Stiefel manifold V_k,m and the manifold P_k,m−k of m×m orthogonal projection matrices idempotent of rank k which is equivalent to the Grassmann manifold G_k,m−k. Asymptotic theorems are derived when the concentration parameters of the distributions are large. We investigate the asymptotic behavior of distributions of some (matrix) statistics constructed based on the sample mean matrices in connection with testing hypotheses of the orientation parameters, and obtain asymptotic results in the estimation of large concentration parameters and in the classification of the matrix Langevin distributions.     

Varieties generated by lattices of breadth two

Some relations between the classB of lattices of breadth at most two and its subclassD of dismantlable lattices, as well as the lattice varietiesV (B) andV (D) generated byV (D) andV (D), respectively, are studied in this paper. For finite join-semidistributive lattices, the two concepts of dismantlability and breadth at most two coincide. There are infinitely many lattice varieties between the varietiesV (D) andV (B), none of them is finitely based.     

Shrinkage estimators of the location parameter for certain spherically symmetric distributions

We consider estimation of a location vector for particular subclasses of spherically symmetric distributions in the presence of a known or unknown scale parameter. Specifically, for these spherically symmetric distributions we obtain slightly more general conditions and larger classes of estimators than Brandwein and Strawderman (1991,Ann. Statist.,19, 1639–1650) under which estimators of the formX +ag(X) dominateX for quadratic loss, concave functions of quadratic loss and general quadratic loss.Research supported by NSF grant DMS-88-22622     

Multivariate nonparametric tests for independence

Multivariate generalizations of Bhuchongkul's bivariate rank statistics [Ann. Math. Statist.35 (1964)] have been introduced and studied in this paper for the purpose of testing mulitvariate independence. It is shown that the test statistics can be expressed as rank statistics which are easy to compute, have asymptotic normal distributions, and can detect mutual dependence in alternatives which are pairwise independent. The tests are compared to the Puri-Sen-Gokhale [[8]] tests and a normal theory test [ [1]] using Pitman efficiency.     

Un Algorithme pour le Calcul des Syzygies sur V[X] dans le cas où V est un Domaine de Valuation

Soit V un domaine de valuation. Nous donnons un algorithme pour calculer une base du V-saturé d'un sous-module de type fini d'un V-module libre (avec une base éventuellement infinie). Nous l'appliquons pour calculer le V-saturé d'un sous-V[X]-module de type fini de V[X] ⁿ (n ∈ ?*). Ceci permet enfin de calculer un système générateur fini pour les syzygies sur V[X] d'une famille finie de vecteurs de V[X] ^k .

We give an algorithm for computing the V-saturation of any finitely generated submodule of V[X] ⁿ (n ∈ ?*), where V is a valuation domain. This allows us to compute a finite system of generators for the syzygy module of any finitely generated submodule of V[X] ^k .     

Characterization of the oblique projector with application to constrained least squares

We provide a full characterization of the oblique projector U(VU)^†V in the general case where the range of U and the null space of V are not complementary subspaces. We discuss the new result in the context of constrained least squares minimization which finds many applications in engineering and statistics.     

Algebras with a compatible uniformity

Given a variety of algebras V, we study categories of algebras in V with a compatible structure of uniform space. The lattice of compatible uniformities of an algebra, Unif A, can be considered a generalization of the lattice of congruences Con A. Mal'cev properties of V influence the structure of Unif A, much as they do that of Con A. The category V[CHUnif] of complete, Hausdor. such algebras in the variety V is particularly interesting; it has a factorization system , and V embeds into V[CHUnif] in such a way that is the subcategory of onto and the subcategory of one-one homomorphisms. Received February 17, 2000; accepted in final form April 1, 2001.     

Some distribution theory relating to confidence regions in multivariate calibration

Summary In the problem of multivariate calibration, Williams (1959,Regression Analysis, Wiley) and Wood (1982, to appear inProc. 11th Internat. Bio. Conf.) have proposed a decomposition of the usual Hotelling'sT ² statistic into the sum of two statistics for use in constructing confidence regions. This paper presents general results for the moment terms basic to Fujikoshi and Nishii's (1984,Hiroshima J. Math.,14, 215–225) approach to the distributions of these statistics, and presents simple alternative approximations to their percentiles.     

New global optima results for the Kauffman NK model: handling dependency

The Kauffman NK model has been used in theoretical biology, physics and business organizations to model complex systems with interacting components. This paper presents new global optima results for the NK model by developing tools for handling dependency in the cases where K grows with N; this generalizes the previous work that focused on the analysis of the (independent) case K=N−1. A dependency graph is defined and studied to handle dependencies among underlying random variables in the NK model. Order statistics (with dependencies) and the expected value of the global optima, E _{N, K} , are bounded using equitable coloring of the dependency graph. These bounds convert the problem of bounding order statistics of dependent random variables into that of independent random variables while incorporating quantitative information about the mutual dependencies between the underlying random variables. An alternative upper bound on E _{N, K} using direct arguments is also proposed. A detailed analysis of E _{N, K} for K close to N (K=N−α and K=β N, α ∈ Z ⁺,β ∈ (0,1)) is given for underlying uniform and normal distributions. Finally, for bounded underlying distributions, the global optima is shown to be concentrated around its mean E _{N, K} .     

One Setting for All: Metric, Topology, Uniformity, Approach Structure

For a complete lattice V which, as a category, is monoidal closed, and for a suitable Set-monad T we consider (T,V)-algebras and introduce (T,V)-proalgebras, in generalization of Lawvere's presentation of metric spaces and Barr's presentation of topological spaces. In this lax-algebraic setting, uniform spaces appear as proalgebras. Since the corresponding categories behave functorially both in T and in V, one establishes a network of functors at the general level which describe the basic connections between the structures mentioned by the title. Categories of (T,V)-algebras and of (T,V)-proalgebras turn out to be topological over Set.